MacMahon's Ω_≥ operator: A computational framework

2025/12/17 by Liu, Feihu, Xin, Guoce
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.15355

Abstract

MacMahon introduced partition analysis in his book ``Combinatory Analysis'' as a computational technique for solving problems related to systems of linear Diophantine equations and inequalities. This paper aims to develop a fundamental computational method for MacMahon's partition analysis. As applications, we present simplified computations for ``Han's formula'', the ``k-gon partitions problem'', and the ``two-dimensional problem''. Moreover, we apply our method to solve a challenging problem.

Citations

• On P-partitions Extended by Two-Rowed Plane Partitions
• MacMahon's partition analysis XIII: Schmidt type partitions and modular forms
• Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems
• MacMahon's partition analysis XII: Plane partitions
• A Residue Theorem for Malcev-Neumann Series
• A residue theorem for Malcev–Neumann series
• A Fast Algorithm for MacMahon's Partition Analysis
• A General Algorithm for the MacMahon Omega Operator
• MacMahon's Partition Analysis VI: A New Reduction Algorithm
• Macmahon's partition analysis IX:K-gon partitions
• MacMahon’s Partition Analysis: The Omega Package
• MacMahon's Partition Analysis
• MacMahon's Partition Analysis: II Fundamental Theorems
• The Sixtieth William Lowell Putnam Mathematical Competition

Related

• Macmahon's partition analysis IX:K-gon partitions
• MacMahon's partition analysis XII: Plane partitions
• Partition Analysis and Symmetrizing Operators
• A Fast Algorithm for MacMahon's Partition Analysis
• Five Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems